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\begin{document}

\centerline{\huge
\textbf{Complex Map} $z \mapsto z^2$}
\bigskip
\centerline{(\textit{From the 3D-XplorMath Project, \url{http://3d-xplormath.org}})}

%  Section headings and comments below are suggestions for further discussion...
%\heading{About This Exhibit}


%  BASIC DESCRIPTION OF THE MATHEMATICAL OBJECT GOES HERE.
%  (Eg. What is the object?  What equations defines it?  Why is it interesting?)

\Large{
See the discussion in ``About this Category'' for details on what to look at, what to expect, and what to do.
We recall that when an example is selected in the Conformal Map Category, first a grid is shown
and then the image of that grid under the conformal map.
The first conformal map examples to look at, (using Cartesian  {\bf and} Polar
Grids) are $ z\to z^2,\ z\to 1/z,\ z\to \sqrt{z},\ z\to e^z$.
}

%  Discussion of what is shown on the screen, parameters, default morphs.
\Large{
REMARK: The actual mapping for this example is 
$z \mapsto aa (z - bb)^{ee} + cc$, 
% THE PARAMETERS OF THIS EXHIBIT:
%    umin:  Real.  Default value: 0.05.  
%    umax:  Real.  Default value: 1.25. 
%    vmin:  Real.  Default value: -1.6.  
%    vmax:  Real.  Default value: 1.6.
%    U Resolution:  Integer.  Default value: 6.
%    V Resolution:  Integer.  Default value: 18.
with default values 
$aa = 1$, $bb = 0$, $cc = 0$, and $ee = 2$. The default morph goes from $ee = 1$
(which gives the identity map) to $ee = 2$ (which gives the squaring map).
}

\vskip 20pt
\huge

%\heading{More About the Math}
%  More advanced mathematical treatment and other materials could go here.

\LF
Just as the appearance of the graph of a real-valued function
$x \mapsto f(x)$ is dominated by the critical points of $f$, 
it is an important fact that, similarly for a conformal map, 
$z \mapsto f(z)$, the overall appearance 
of an image grid is very much dominated by those points $z$ 
where the derivative $f'$ vanishes. 
Most obviously, near points $a$ with $f'(a)=0$
the grid meshes get very small and, as a consequence, the grid lines
usually are strongly curved. If one looks more closely then one notices
that the angle between curves through $a$ is {\bf not} the same as the
angle between the image curves through $f(a)$ (recall: $f'(a)=0$).
We will find this general description applicable to many examples.
\LF
The behaviour of the simple quadratic
function $z\to z^2$ near $a=0$, both in Cartesian and in Polar
coordinates, repeats itsself so closely with other functions, that it is
strongly recommended to play with this function for a while. 
One sees that a rectangle, which touches $a=0$ from
one side is folded around $0$ with strongly curved parameter lines,
and one also sees in Polar coordinates
that the angle between rays from $0$ gets {\bf doubled}. The image
grid in the Cartesian case consists of two families of orthogonally
intersecting parabolas. One may use the Action Menu and add a
circle that passes through the origin in the domain grid. The program 
maps it to the well known, apple shaped, cardioid in the range grid.
\LF
One should return to this prototype picture after one has seen others
and looked at the behaviour near their critical points. Good examples 
are  $z\to z+1/z$, $z\to z^2 + 2z$, $z\to \sin z$ and even the Elliptic functions.

\author{H.K.}

\end{document}


%  Section headings and comments below are suggestions for further discussion...


%\heading{About This Exhibit}
%  Discussion of what is shown on the screen, parameters, default morphs.

%\heading{Things to Try}
%  Suggestions for other things that can be tried with the exhbit.



%\heading{More Information}
%\Large
%  Links to material on the Internet could go here
%  Example:  See \href{http://3d-xplormath.org}{the 3D-XplorMath web site}

\end{document}


% THE PARAMETERS OF THIS EXHIBIT:

%    b:  Integer.  Default value: 2.
%    Coefficient a of z^b:  Complex.  Default value: 1.0.  Default morph from 0.0 to 1.0.
%            Hint: The map is: z --> b*z + a*z^b 
%    umin:  Real.  Default value: -1.0.  Default morph from -1.0 to -1.0.
%    umax:  Real.  Default value: 1.0.  Default morph from 1.0 to 1.0.
%    vmin:  Real.  Default value: 0.0.  Default morph from 0.0 to 0.0.
%    vmax:  Real.  Default value: 2*pi.  Default morph from 2*pi to 2*pi.
%    U Resolution:  Integer.  Default value: 10.
%    V Resolution:  Integer.  Default value: 31.

